Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  abssf Structured version   Visualization version   GIF version

Theorem abssf 39813
 Description: Class abstraction in a subclass relationship. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypothesis
Ref Expression
abssf.1 𝑥𝐴
Assertion
Ref Expression
abssf ({𝑥𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))

Proof of Theorem abssf
StepHypRef Expression
1 abssf.1 . . . 4 𝑥𝐴
21abid2f 2930 . . 3 {𝑥𝑥𝐴} = 𝐴
32sseq2i 3772 . 2 ({𝑥𝜑} ⊆ {𝑥𝑥𝐴} ↔ {𝑥𝜑} ⊆ 𝐴)
4 ss2ab 3812 . 2 ({𝑥𝜑} ⊆ {𝑥𝑥𝐴} ↔ ∀𝑥(𝜑𝑥𝐴))
53, 4bitr3i 266 1 ({𝑥𝜑} ⊆ 𝐴 ↔ ∀𝑥(𝜑𝑥𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1630   ∈ wcel 2140  {cab 2747  Ⅎwnfc 2890   ⊆ wss 3716 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-in 3723  df-ss 3730 This theorem is referenced by:  rabssf  39820
 Copyright terms: Public domain W3C validator